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积分应用：相交圆柱体体积与铲雪车问题

Applications of Integration

专题
General / 综合
难度
L4

题目详情

  1. 两根半径为 1 的无限长圆柱互相垂直相交（轴线正交且相交）。求交集体积。

  2. 早晨某时刻开始下雪，降雪强度恒定。中午（记 t=0t=0）开始以恒定“体积/小时”清雪。到 t=1t=1（下午 1 点）清了 2 英里，到 t=2t=2 清了 3 英里。问：几点开始下雪？

Two cylinders of radius 1, intersecting perpendicularly with centers crossing. Find the volume of the intersection.

Suppose it starts snowing at some unknown time in the morning, at a constant rate. At noon (t=0t=0), a snowplow starts clearing the road at a constant volume rate per hour. At t=1t=1 pm, it has gone 2 miles; at t=2t=2 pm, it has gone 3 miles. When did it start snowing?

解析
  1. 交集体积（Steinmetz solid）：
V=163.V=\frac{16}{3}.
  1. 设中午前 TT 小时开始下雪，则雪深与时间成正比，清雪速度与雪深成反比，可写为 v(t)=ct+Tv(t)=\frac{c}{t+T}。

由路程积分：

∫01ct+Tdt=2,∫02ct+Tdt=3.\int_0^1\frac{c}{t+T}dt=2,\quad \int_0^2\frac{c}{t+T}dt=3.

联立解得

T=5−12.T=\frac{\sqrt5-1}{2}.

即中午前 5−12≈0.618\frac{\sqrt5-1}{2}\approx 0.618 小时开始下雪（约 11:23）。

Original Explanation

The result is Vintersection=163.V_{\mathrm{intersection}} = \frac{16}{3}.

Let T>0T>0 be the hours before noon when snow began. Snow depth (cross-sectional area) A(t)=c2 (t+T).A(t)=c_2\,(t+T). The plow's speed  v(t)=c1A(t)=ct+T.\,v(t)=\tfrac{c_1}{A(t)}=\tfrac{c}{t+T}.

Hence ∫01ct+T dt=c ln⁡(1+T T )=2,∫02ct+T dt=c ln⁡(2+T T )=3.\int_0^1 \frac{c}{t+T}\,dt = c\,\ln\Bigl(\frac{1+T}{\,T\,}\Bigr)=2, \quad \int_0^2 \frac{c}{t+T}\,dt = c\,\ln\Bigl(\frac{2+T}{\,T\,}\Bigr)=3. Therefore ln⁡(1+TT)3=ln⁡(2+TT)2  ⟹  (1+TT)3=(2+TT)2.\ln\Bigl(\frac{1+T}{T}\Bigr)^3 = \ln\Bigl(\frac{2+T}{T}\Bigr)^2 \;\Longrightarrow\; \Bigl(\frac{1+T}{T}\Bigr)^3 = \Bigl(\frac{2+T}{T}\Bigr)^2. Solving gives T=5−12.T = \frac{\sqrt{5}-1}{2}.